LOGICAL CONSEQUENCE IN MODAL LOGIC: NATURAL DEDUCTION IN S5 by JOHN CORCORAN and GEORGE WEAVER

370

Notre Dame Journal of Formal LogicVolume X, Number 4, October 1969

LOGICAL CONSEQUENCE IN MODAL LOGIC:NATURAL DEDUCTION IN S5

JOHN CORCORAN and GEORGE WEAVER

This paper presents a (modal, sentential) logic -£DD which may bethought of as a partial systematization of the semantic and deductiveproperties of a sentence operator (D) which expresses certain kinds ofnecessity. The language of .DD is LDD, the smallest set of formulascontaining a countably infinite set of sentential constants and closed underforming Πφ, ~φ, and (φ z> ψ) from any formulas φ and ψ already in the set.This is essentially the language of S5. The semantics of -£DD is essen-tially Kripke's semantics for S5. In the semantics for .£DD, however, aninterpretation of LDD is defined as an ordered pair (a, P) where a is anordinary truth-value interpretation of the sentential constants and P is aset of such interpretations with a ε P. (α, P) is a model of φ if φ is trueunder (α, P) and {a, P) is a model of S c LDD if each ψ in S is true under(a,P). This permits logical consequence to be defined: for S c LDD,φε LDD, φ is a logical consequence of S(S\=φ) iff every model of S is amodel of φ. The deductive system consists of "natural" rules permittingproofs from arbitrary sets of premises and we let S\-φ mean that there isa proof of φ from the premises S.

Let LD be the sublanguage of LDD containing all formulas of LDD de-void of iterated or nested D. Let -£D be the restriction of ^DD to LD, i.e.£π is the logic with language LD such that, for S <Ξ LD and φε LD, S \=φrelative to -£D iff S 1= φ in «£DD and S hφ relative to -£D iff there is aproof in -£DD of φ from S containing only formulas of LD. Strong com-pleteness (S t= φ implies S \-φ) for -£DD is proved from the following threelemmas: (1) strong soundness (Sϊr-φ implies S t= φ) for -£DD, (2) strongcompleteness for -£D and (3), a reduction theorem to the effect that everyformula in LDD is provably equivalent in -£DD to a formula in LD. Fromthese results, together with Kripke's weak soundness and weak complete-ness results for S5, it follows that φ is a theorem of S5 iff φ is a theoremOf ZDD.

1 The Logic of -£DD. Let O be a countably infinite set of symbols calledsentential constants and let LDD be the smallest set of formulas containing

Received August 16, 1968

LOGICAL CONSEQUENCE IN MODAL LOGIC 371

& and closed under forming Πφ, ~φ and (φ z> ψ) from any formulas φ andi// already in the set. The set of (purely) modal formulas is the smallestcontaining all formulas in LΠΠ of the form Πφ and closed under forming~ψ and (ψ D η), Let L be the sentential language included in LDD, i.e. L isthe smallest set of formulas containing & and closed under forming ~ψ and(Ψ 3 v).

1.1 The interpretations of LDD are to be ordered pairs {a, P) where a isan ordinary truth-value interpretation of L and P is a set of such interpre-tations with aεP. In a particular interpretation (a, P) ,a is to be thought ofas an "actual world" and P is to be thought of as a set of "possibleworlds". We use the notation aP to indicate an ordered pair, (a,P).Accordingly, formulas in L are true in aP iff they are true inβrformulasnot involving necessity (D) have their truth-values determined by theactual world a. Πφ will be defined as true in aP iff φ is true in all bP,bεP. Thus, in effect we will have, in addition to the actual world a and theset of possible worlds P (both of the interpretation aP), the "actual"interpretation aP and its corresponding set of possible interpretations:{bP:bεP}. The corresponding set of possible interpretations for a possibleinterpretation is the same as that of the actual interpretation. The aboveremarks might aid the reader in seeing beyond the formal developmentwhich follows.

Below a, b, a\ b\ etc. are functions from & to {t, f} and P, P\ etc. aresets of such functions. aP is an interpretation (of LDD) iff aεP. For eachinterpretation aP we define a unique function VaP from LDD to {t, f} in sucha way that for each φε LDD, VaP(φ) = t(= f) when and only when φ is to beregarded as true in aP (false in aP). Let N and C be the usual truth-functions corresponding to ~ and D.

Definition: For φ ε&, VaP(φ) =a(φ). If φ = ~ψ then VaP(φ) =N(VaP{ψ)). Ifφ = (ψ D η) then VaP{φ) = C(VaP(ψ), VaP{η)). If φ = Πψ then VaP(φ) = t iffVbP(ψ) = t, for all bεP. V"pis called the (truth) valuation in aP.

If VaP(φ) = t then aP is a model of φ. If aP is a model of every formulain S then aP is a model of S. If every interpretation is a model of φ then wesay that φ is logically true and write \=φ. If every model of 5 is a model ofφ, we say that φ is a logical consequence of S and write S \=φ. Thus, lettingΛ denote the empty set, A\=φ iff \=φ. For conciseness we write TS, φr

instead of τSu{<rf-τ and "φ \=ψ' instead of }{φ}t=ψτ and accordingly, thefollowing are consequences of the above definitions.

50.0 If S\=φ, for all φεS\ and Sτ N ψ, then S\=ψ.50.1 If S c Sτ and S \=φ then Sτ f= φ.51 S, φ\=φ.52 If S\=φ and S\=(φ z> ψ) then S\=ψ.53 US, φ\=ψ then S\=(φ D ψ).54 If S, -<^(=ψ- and S, ~<p μ~ / then SN^.55 If S\=Πφ thenS\=φ.56 If all formulas in S are modal and S h<ρ then S ND(p.

372 JOHN CORCORAN and GEORGE WEAVER

The first three of these principles follow from the definitions of "\="and "model of S" without regard for the particular character of thesemantics. The next three hold because the valuations preserve thenormal meanings of ~ and D. S5 follows since VaP(Πφ) = t impliesVaP(φ) = t.

S6 follows from the following proposition: If φ is a modal sentencethen for all a, bεP, VaP(φ) =VbP{φ). To see this, assume the propositionand let 5 be a set of modal formulas such that S \=φ. Let aP be any modelof S. If S is empty then φ is true in all interpretations, in particular in allinterpretations bP with bεP. Thus VaP{Πφ) = t. If S is not empty then, bythe proposition, bP is also a model of S for any bεP. Since S (= φ each suchinterpretation must be a model of φ. Thus VaP(Πφ) = t. Since aP is anymodel of S, S \=Πφ.

The proposition mentioned above is proved as follows. Let P and a bearbitrary except that aεP. Consider the property of formulas φ: for allbεP, VaP(φ) = VbP{φ). From the definition of valuation it follows that theproperty holds for all formulas of the form Πψ and also that it holds for~ψ and (ψ D η) whenever it holds for ψ and η. Thus it holds for all modalformulas.

1.2 The deductive system for -£DD parallels the semantic principles (SO.Othrough S6) so closely that the (strong) soundness proof is quite simple.The sentential (or tautologous) part of the system is essentially Jaskowski'ssystem (referred to by Leblanc [7], p. 16). This consists of the familiarrules: assumption (A), repetition (R), detachment or modus ponens (i^E),conditionalization (DI) and a form of reductio ad absurdum (~E). The modalrules are G introduction (DI) and D elimination (DE).

The proofs are written left-to-right on one line rather than top-to-bottom on several lines. Brackets, ] and [, are used as punctuation in theproofs. Thus proofs will be strings over LDDU{], [}; i.e. certain strings offormulas of LDΠ having ] and [ interposed as punctuation will be proofs.An occurrence of [φ at a place in a proof Π indicates that φ is assumed inΠ at that place. An occurrence of ]φ at a place in a proof Π indicates(a) that the right-most assumption (not already discharged) is dischargedand (b) that φ is derived from the remaining (undischarged) assumptions.An occurrence of φ (not preceded by ] or [) at a place in a proof indicatesthat φ is derived from the undischarged assumptions preceeding it. Inorder to give the reader an orientation to the new linear notation, we giveseveral familiar natural-deduction proofs using it.

Examples'. The string (a) below is a proof of (φ z> φ) from Λ, (b) is a proofof (φ D (ψ Z) φ)) from Λ, (c) is a proof of ((φ 13 ψ) D (φ D η)) from {φ z>{•ψ D η)), (d) is a proof of φ from (~φ Ώ ~ψ) and (φ 3 ψ) and (e) is a proofof ~φ from (φ D ψ) and (φ ZD ~ψ).

(a) [φφ] (φ D φ)

(b) [φ[ψφ](ψ D φ)](φ^> (Ψ =><?))(c) [(φ D (ψ D η)) [(φ D ψ) [φψ(ψ D η) η] (φ D η)] {{φ D Ψ) 3 (φ D η))(d) [(~φΏψ)[{~φΏ~ψ)[~φψ~ψ]φ

(e) [(φ D ψ)[(φ D ~ψ)[—φ [~φ~φ—φ]φΨ ~ψ]~φ


Proof (e) has more lines than necessary but the superfluous lines makethe application of (~E) more transparent.

We say that an occurrence of a formula in a proof is open if it isderived from undischarged assumptions and that it is closed if it is derivedfrom discharged assumptions. Because the concepts of open and closed arecrucial in discussing proofs we will give fairly rigorous definitions of themdespite the fact that "closed" amounts to being enclosed in paired bracketsand "open" is simply "not closed". It is also perhaps desirable from thepoint of view of exposition to first define a fairly "small" set of stringswhich will include all proofs.

Proof expression (p.e.) is defined recursively as follows: [φ is a proofexpression, if Π is a proof expression so are Π [φ, Uφ and Il]φ where < isany formula of LDD. No string (over LDD {], [} is a p.e. except by theabove two rules. Proofs will be defined as certain of the proof expressions.E.g., [φφ will be a proof and [φ~φ will not.

Now we will define "closed in a p.e." (enclosed between pairedbrackets) and "open in a p.e." recursively by reference to substrings ofproof expressions called closed sections. Closed section is defined asfollows: [φiφ2, . , φn] is a closed section. If β is a closed section so is[<?i<?2, , ψnβΨiΨ2, . , Ψn] No string (over LDD {],[}) is a closedsection except by the above two rules. Let β be a p.e. An occurrence of φin β is closed in β iff it is within a closed section which is a substring of β.An occurrence of φ is open in β iff it is not closed in β. An occurrence of aformula in β which is immediately preceded by a [ is an assumption in β. Aformula (not an occurrence) is an assumption of β iff it occurs as an openassumption in β.

Definition of Proof: In the following rules φ, ψ and η are formulas ofLDD; Π, Πi, and σ, στ are strings over LDDU{], [}. Π is a proof iff Π isconstructed by a finite number of applications of the following rules.

(A) [φ is a proof and if Π is a proof then Il[φ is a proof(R) If Π is a proof and φ occurs open in Π, then Uφ is a proof.

(DE) If Π is a proof, φ occurs open in Π and (φ ZD ψ) occurs open in Πthen Uψ is a proof.

(i3l) If Π = σ [φσ'ψ is a proof where the indicated occurrence of φ is theright-most open assumption in Π then Il]{φ D ψ) is a proof.

(~E) If Π = σ[~φσ' is a proof where (1) the indicated occurrence of ~φis the right-most open assumption in Π and (2) both ψ and ~ψoccur open in Π, for some ψ, then Il]φ is a proof

(DE) If Π is a proof and Uφ is open in Π then Uφ is a proof.(D I) If Π is a proof, φ occurs open in Π and all open assumptions in Π

are modal formulas then UΠφ is a proof.

Notice that in (DI) and (~E), if σ is not null then σ is a proof. More-over, every proof Π ends with an open formula. The last formula in Π iscalled the conclusion of Π and is denoted by C(Π). The set of (open)assumptions of Π is denoted by A(Π). We define: S h φ iff there exists a Π


such that A(Π) <Ξ s and C(Π) = φ. Instead of Λ v φ we write \-φ. If \-φ, φ issaid to be a theorem (of -£DD).

Before considering the soundness of / D D it is appropriate to demon-strate some facts about the deductive system alone. By suitable extensionsof proofs given above we have:

D0.1 (φ Z) {ψ Ώ φ))

D0.2 ((φ D (ψ D 77)) D ((φ D ψ) D (φ z> 77))

i)Λ3 ((-<? D ψ) D ((-<? D~ψ) D <?))D0.4 (φ D <p)

Z>6>.5 ((φ D ψ ) D ((<p D ~ψ) z> -<^))

As meta-proofs below we either exhibit an object-proof or indicate howan object-proof may be constructed.

DO, 6 (φ 3 — φ )[φ[~—φ[—φ]~φ]~~φ](φ D — φ )

DO. 7 ( — φ D φ)

[—φ\rφ]φ]{—φ ^ φ)D0.8 {{φ Dψ)D ({~φ D ψ) 3 ψ))

Let us verify the obvious fact that

Z)βO IfS \-φ and S H (φ D 1//) ίftew S h-ψ

Proof: Assume S H<ρ and SH(<^ D ψ). Thus there are two proofs, say Πand ΓΓ, such that A(Π) U A(ΠT) c s, C(Π) = φ and C(ΠT) = (<p D ψ). Π Πτ isalso a proof with A(Π Πτ) ζ S . Moreover, ^ and (<ρ D ψ) are both open inΠ Πτ. Thus Π Jl'ψ is a proof with A(Π U'ψ) Q S and C(Π Wψ) = ψ.

Now we consider some facts involving D.

Dl.l \-{Πφ D φ)

[Uφφ]{Ώφ Ώ φ)D1.2 \-(~nφ D D D(/?)

Z)ί.3 h(D(φ D ψ) D (D< 3 ΠΨ))[D(<p D ψ ) [ D ^ D ι//)ψDi//] ( α ^ 3Dψ)](D(ςp D ψ ) D (Dςp DDψ))

D2.4 h(D<p DDD(^)[D </? DD φ](Πφ D DD φ)

i)2.5 l-(<p D ~D^(^)[(^ [^"^ D ~φ \y Ώ~φ ]π~φ~φ ]~Ώ~φ ](φ D ~Π~φ)

D1.6 \-(φ D D~D~<p)

To the end of the previous proof adjoin the following.

The indirectness of the proof of D1.6 indicates a quite far-reachingtroublesome situation. Specifically: there are infinitely many non-modalformulas φ such that φ \=Πψ for some ψ, but Πψ can not be proved directly


from φ by proving ψ from φ and then boxing. (Dl) can only be used when allopen assumptions are modal. This restriction is obviously necessary.Thus if -£DD is to be strongly complete, there must always be an"indirect" way of getting from such a φ to such aDψ.

DR1 Ifhφand\-(φZ) ψ)then\-ψ

Proof: Special case of DRO when S = Λ.

DR2 Ifhφ then \-Uφ

Proof: Let Π be a proof of φ from Λ. All open assumptions are modalformulas (vacuously). Thus WUφ is a proof of Uφ from Λ.

1.3 In this section we compare the deductive system of JLΠΏ both with aformulation of sentential logic (Mendelson [8], pp. 30-37) and with a formu-lation of S5 as a logistic system (Kripke [6], pp. 67-68).

Let SCx be the following sentential logic. L, the language of SCi, is theset of formulas of LDD devoid of D. An interpretation, α, of L is a functionfrom β> to {t, f}. As usual, the truth-valuation in a is the function Va fromL to {t, f} such that Va{φ) = a(φ) for φ εύ>, Vβ(~<p) = NVa(φ), Va{(φ D ψ)) =C(Va(φ), Va{ψ)). a is a model of φ iff Va{φ) = t. a is a model of 5 c L iffVa(φ) = t for all φ εS. S T N φ (S tautologically implies φ) iff every model ofS is a model of φ. If Λ T (= φ then we write T (= φ and call φ a tautology asusual. The deductive system of SCX is defined as follows: Π is a proof inSCX iff Π is a proof in /DD and every formula in Π is in L. ST \-φ iff thereis a proof Π in SCX such that A(Π) <^S and C(Π) = φ. If Λ T \- φ then wewrite Ύ \-φ and say that φ is tautologically provable.

By considering the proofs of D.01, DO.2, D0.3 and DRO it is clear thatthey still hold when t- is replaced by TK It follows by the reasoning inMendelson already cited that SCi is complete, i.e., that T \=φ impliesT\-φ. Thus the sentential rules of -ώUϋ form a complete sentential logic.

S Q L is consistent (in the sense of SCi) iff there is no ψε L such thatSTHψ and S T t-~ψ. S Q L is maximally consistent (in the sense SC^ iff Sis consistent and, for φεL, if φjίS then S, φ is inconsistent. By the rea-soning in Davis [3], pp. 7-9, every consistent set is a subset of a maximallyconsistent set and every maximally consistent set S has a unique model(viz. for each φεσ, a{φ) = t if φεS and a{φ) = ί if φfίS). The latter factwill be used below. It follows that SCX is strongly complete {ST\=φ impliesSΎh-φ): suppose S T |= φ; then S, ~φ has no model; by the above result,S, ~φ is inconsistent; S,~φT)rψ and S,~φ T \-~ψ; using (-Έ), ST \-φ.

Let SC2 be exactly like SCX except that the language is now all of LCD.Interpretations will be functions from &u{Πφ:φε LDD} to {t, f} and theproofs are those of -£DD not involving use of (DE) or (Dl). All of the aboveresults for SCX hold mutatis mutandis for SC2. Thus all of the tautologiesin LDD are theorems of SC2.

Except for a change in notation, the set of theorems of S 5 as formu-lated by Kripke (loc. cit.) is the smallest set containing all tautologies ofLDD and all formulas of the forms Dl.l, D1.2 2SΛ D1.3 and satisfying DR1


and DR2. Thus if ςί> is a theorem of S5 then φ is a theorem of ^.DD.Kripke showed that S5 is complete relative to the semantics of -£DD, i.e.,if \=φ then φ is a theorem of S5. It follows then that

Theorem 1.3: .£•• is {weakly) complete in the sense that \=φ implies \-φ.

1.4 The completeness theorem for ^DD and other provability results, aswell, are without significance unless ^DD is at least shown to be con-sistent (i.e. there is no φ such that y-φ and i—φ). Moreover, the deductivesystem of JLUΠ is inadequate unless every one of its proofs Π can be reliedon to establish logical consequence between premises and conclusion(A(Π)NC(Π)). In this section we demonstrate that -£DD has this property.It will follow that j£DD is strongly sound, i.e. that S\-φ implies S\=φ.From strong soundness it follows that -£DD is (weakly) sound (\-φ implies\=φ) and from (weak) soundness consistency follows.

Lemma 1.4: For every proof Π in -£DD, if φ occurs open in Π then

A(Π)h<ρ.

Proof: The proof is by a "course-of-values" induction on the proofs of-£DD. The lemma is obvious for proofs of the form [ψ. Now let Π be anarbitrary proof and assume that the lemma holds for all proper subproofsof Π, i.e. assume that for all ΓΓ such that Π = IΓσ, σ non-null, A(IΓ) \=φ forall φ open in IT. This is the induction hypothesis. We show that the lemmaholds for Π. There are seven cases according to which of the seven rulesof inference was used to form the last "s tep" of Π. In carrying out theseseven cases we had found it convenient to define for each Π the set offormulas occurring open in Π. Since we merely sketch the proof here, wewill not introduce any new notation for this concept.

In case (A) was the last rule used, the induction step follows from theinduction hypothesis, S0.1 and SI. For (R) as the last rule used, the induc-tion step follows from the induction hypotheses. For (DE), (DI), (~E), (DE)and (DI) the induction step follows by the induction hypothesis, SO.0 and,respectively, S2, S3, S4, S5 and S6. The induction step follows in general,therefore, and the proof is complete.

Incidentally, since the deductive systems ofSCi, SC2 and the sententialpart of ZΠD are all included in that of ZΠD and since S0.0, S0.1, SI, S2,S3, S4 and S5 all hold for the semantics of sentential logic; it follows by theabove reasoning that the theorem holds for these logics as well.

Theorem 1.4: -£DD is strongly sound, i.e. S \-φ implies S \=φ.

Proof: Assume S \-φ. Then there exists a proof Π with A(Π) Q S and C(Π) =φ. Since φ is open in Π, by the lemma A(Π) |=<ρ. By S0.1, S\=φ.

In the special case where S is empty we have the weak soundnessresult for ^DΠ, i.e. \-φ implies \=φ. Putting this together with Theorem1.3 (weak completeness) we have what is called by some logicians a "weakbridge result" for «ZΠD, viz. \-φ iff \=φy thereby partially bridging thesemantics (N) and the deductive system (H). Since S5 (as formulated byKripke) also has the weak bridge property we have that φ is a theorem ofS5 iff φ is a theorem of ZΠD.


Note: The rule (R) is superfluous in the sense that if S\-φ in .DD thenthere is a proof in -£DD of φ from S which does not make use of (R). Thisis proved using the following proposition: if Π is a proof in -£DD and IT isthe result of deleting from Π the first occurrence of a formula obtained byuse of (R) then Πτ is a proof in /DΠ and every φ open in Π is open in IT.Now assume S\-φ. It follows that there is a proof Πo of φ from S such thatevery other proof of φ from S contains at least as many formulas as Π, i.e.it follows that there is a "shortest" proof Πo of φ from S. By the above-mentioned proposition Πo cannot contain any formulas obtained by use of(R).

2 The Logic £π Here we consider a proper subsystem JLΠ of -£DD. Thelanguage of . £ • is LD, the set of formulas of LDP without iterated D. Moreprecisely, LD is the smallest set of formulas (1) containing all formulas inL, (2) containing all formulas Πφ with φ in L and (3) closed under forming~φ and (φ D ψ) from φ and ψ already in the set. The semantics of JCΠ isexactly the same as that of ZDD except all valuations VaP are restricted toLD. The proofs of JLΠ are the proofs of /DD which contain only sentencesin LD—i.e. the only changes in the rules come (1) for (DI) where one cannotform Πφ unless φ has no D occurring in it and (2) for (A) where one cannotwrite [φ unless φ is in LD. Here we use N and \- relative to £π. If thereis any chance of confusion, we will add "in «£D" or "in «£DD'\

Theorem 2.1: JLΠ is strongly sound.

Proof: Let S, φ c LD. Suppose S \-φ in ZD. Then S \- φ in ^DD. ByTheorem 1.4 (strong soundness), S \= φ in -£DD. But since S, φ c LD, itfollows by the remarks of the last paragraph that S \=φ in JLπ.

Our object now is to show strong completeness for JLΠ, i.e. that S\=φimplies S\-φ. The proof parallels the ideas of Henkin [5]. Consistent andmaximally consistent, both relative to £.Π, are defined as above mutatismutandis. By Lindenbaum's argument (Davis [3], p. 7) every consistent setof sentences is a subset of some maximally consistent set. We show that amodel aP can be formed for each maximally consistent set. Thus everyconsistent set of sentences has a model. From this fact, strong complete-ness follows (See section 1.3 above):

First we need some elementary facts about maximally consistent setsof formulas of JiΠ. Let Π(S) = {φ: Πφ ε$

Theorem 2.2 If S is maximally consistent [relative to j£π) then

(1) // φfίS thenS\-~φ,

(2) If S \-φ then φεS9 (Hence φεS iff Shφ),(3) φεS or ~φεS,

(4) -φεSiffφίS,(5) (φ Dψ)εS iff φjiS orψεS,(6) S Π L is maximally consistent in the sense of sentential logic,

(7) D (S) is consistent in the sense of sentential logic, and(8) D(S) \-φ implies φ εΠ{S).


Proof: Parts (1) through (6) are easily proved using D0.1, DO. 5 and thedefinitions. (7) follows from (6) using the observation that Π(S)QS Π L.To see (8), assume that D(S) \-φ. Let Π be a proof of φ from D(S) and letφh φ2, . . . , and φn be the (open) assumptions of Π. Then [Πψx [Uφ2 [Uφz,. . . , [Πφn IT is also a proof where Πτ is obtained from Π by deleting the['s before φh φ2y . . . , and φn in Π—the occurrence of ψi in the new proofis justified by (DE). Since each ψi is in D(S), each Uφ{ is in S. Since allassumptions of the new proof are modal and φ is open in the new proof,Uφ can be added on by (DI). Thus S \-Πφ and by 2, UφεS. Thus φεS.

Next we form a model of any given maximally consistent set. But firstconsider any consistent S <Ξ LD. S Π L is consistent in the sense ofsentential logic and thus has a model a. Thus for anyP withαεP, aP is amodel (in the sense of -£π) of 5 Π L. D(S) is also consistent in the senseof sentential logic and thus has a model, b. LetP τ be the set of all modelsof D(S). For any a1εP\ a'F" is a model of {Πφ:φεΠ(S)}. But aP need notbe a model of S. We will show that in case S is maximally consistent aF"where a is a model of S Π L and P τ is the set of models of D(S) is a modelof S.

Theorem 2.3 If S is maximally consistent and a is the (unique) model ofSO L and P is the set of all models {in the sense of sentential logic) ofD (S) then

(1) for all (pεLD, φεS iffVaP{φ) = tand, therefore,(2) aP is a model of S

Proof: LD is the smallest set containing o and {D<ρ:<pεL} and closed underforming ~φ and (φ z> ψ) for φ, ψ already in the set.

(1) can easily be seen to hold for all formulas φ in O. It also holds forall formulas Πφ for φεL. This is seen as follows. Let φε L. ThenVaP(Πφ) = t iff VaP(φ) = t for all bεP. Since φε L, VbP(φ) = b{φ). ThusVaP(Πφ) = t iff b(φ) = t, for all bε P. Now we show Only if: Suppose Uφε S.Then φε Π(S). Thus b(φ) = t for all bεP (by hypothesis). Thus VaP(Πφ) = t.For 'if we need part (8) of Theorem 2.2. Assume YaP(Πφ) = t. Then b(φ) =t for all bε P. Since P is the set of all models (in the sense of sententiallogic) of D(S), D(S) T \=φ. By strong completeness of sentential logicD (S) v-φ. By part (8) of 2.2, φεΠS. By the definition of Π(S), Πφε S.

It can easily be seen that the property of (1) holds of ~φ and (φ z> ψ) ifit holds for φ and ψ. Thus it holds for all formulas in LD.

Theorem 2.4: JiU is strongly complete.

Corollary 2.4: The semantics of JiU is compact, i.e.

(1) If S \=φ then there is a finite F Q S such that F \=φ and (equivalently)(2) if S has no model then there is a finite F Q S such that F has no model.

Proof: Suppose S \=φ. By strong completeness Shφ. Then there is aproof Π of φ from S. Π can only have a finite set F oϊ assumptions. ThusF h φ. By soundness F \= φ.

Putting Theorems 2A and 1.4 (strong soundness) together we have


S hφ iίί St-φ, the strong bridge result for X.Π. This completely bridgesthe semantics and the deductive system of «£D. In addition, we have thefollowing.

Theorem 2.5: Let N and No, φ Q L and let Mo = {Πφ:φεN0}. Then(1) N\-φ iffNTv-φ(2) \-Πφ iffT\=φ(3) if N is consistent then Nv-Ώφ iff ^Uφ(4) if Mo UN is consistent then Mo U N h- φiff M0\-Πφ(5) M0\-φ iffN0\-φ

Proof: The 'if' parts of (1) through (4) are immediate. We will show the'only if' parts of (1) through (4) first and then show (5). (1): Suppose Nhφ.Then Nϊ=φ. Now we show NT h φ which implies NT \-φ by strong complete-ness of sentential logic. Let a be any model of N. Then a{a} is a model (in-£DD of N. Thus a{a} is a model of φ. Since φεL, a is a model of φ.(2): Suppose \-Πφ. Then \-φ. Hence (=(/?. Let a be any interpretation.Va{a\πφ) = t iff a(φ) = t. But h<ρ. Thus a is a model of φ. So Th<ρ.(3): Suppose Nv-Ώφ where N is consistent. Then N\=Πφ and iV has a modelβ. Hence αP is a model of iV where P is the set of all interpretations of L.Thus VaP(Πφ) = t. But V*p(D<p) = t iff b(φ) = t for all b. Thus T\=φ. ThusT H f So h ? and hD^. (4): Suppose that M0UN is consistent andMo U N\-Πφ. By soundness Mo U N\=^Πφ and there exists a model, #P say,of Mo U iV. We will show Mo \^Uφ from which Mo KDφ follows by strongcompleteness. Let a'P1 be any model of Mo. Then α(P U Pτ) is a model ofMo UN. Why? Suppose ηεN then α(P UP1) is a model of 77 iff a(η) = t.But αP is a model of JV. Suppose ηεM0. Then r] = Πψ and «(P U P !) is amodel of Gψ iff b(ψ) = t for all bεP UP' . But «P is a model of Mo, henceb{ψ) = t for all bεP, and «'PT is a model of Mo, hence b{ψ) = t for all bεP\Hence α(P U Pτ) is a model of Mo U N. Since Mo UiNΓ μn<p, a(F U P') is amodel of Dφ. Thus b{φ) = t for all bε(P U Pτ), in particular, for all bεP\Thus ^TPT is a model of D<ρ. But α τP' was an arbitrary model of Mo; henceM O N D ^ . (5) Since Πψ\-ψ, if NQ\-φ then Mo i-^. Now assume Mo \-φ.Let α be any model of No. a{a} is, therefore, a model of Mo and hence of φ.But φεL so V*UI(<?) = VΛ((/?). Thus N0T Nφ. Thus iV0T h<p. By (1), iV0 h</>.

The significance of these results is as follows. (1) tells us that, if wecan prove a formula of sentential logic from a set of sentential logicassumptions using the system -£D, then we can prove the same thing usingonly the sentential logic rules. Thus (DE) and (Dl) provide a "conserva-tive" extension of sentential logic. (2) tells us that, Πφ is a theorem ofJLπ iff φ is a tautology. (3) tells us that even in the case of a theory, all ofwhose assumptions are devoid of D, Πφ is a theorem of the system iff φ isa tautology. Thus extending SC! to £u enables one to distinguish fromamong the consequences of a set of sentential logic assumptions thosewhich do not depend on the particular assumptions. (4) implies that we canuse -£D to systematize the distinction between assumptions which we mightwant to regard as necessary and other assumptions which we would con-sider merely "factual" or merely true. If we put boxes before the former


and list the later (simply) then: if we can prove Πφ then φ is a consequenceof the necessary assumptions. However, in such a case if we can prove φ,but not Ώφ, it does not follow that φ is a consequence of the "factual"assumptions. Example: DP, P D R HR but R does not follow from P D Ralone. For further discussion along these lines see Curry [2], pp. 359-360(5) tells us that we do not get any "truths" by assuming that our axiomsare necessary that we could not already obtain by taking the assumptions asmerely true.

Note: Notice that, in the proof of strong completeness of ^D, the rule (πl)was used only once, viz. in the proof of clause (8) of Theorem 2.2. Here aweakened rule, (Dlτ), would do just as well.

(DΓ)' If Π is a proof, all of whose assumptions are of the form Πψ and inwhich φ is open, then HΠφ is a proof.

The resulting system is precisely the one called PN in [l]. (GΓ) isanalogous to Curry's sequent rule, [2], p. 361.

It is also of interest to point out that every theorem of ZO is also atheorem of S4, S5 and the Brouwerian System and that every formula in LDwhich is a theorem of S4, S5 or the Brouwerian System is a theorem of/ D . Thus S4, S5 and the Brouwerian System agree on LD. This followsfrom Kripke's soundness and completeness proofs for these systems withthe help of an easy theorem to the effect that for every S4 or Brouwerianinterpretation, i, there exists an interpretation aP such that, for all φε LD,VaP{φ)=Ψ{φ).

3 Strong Completeness of -£DG; The strong completeness of .£GG isproved by reducing the question to that of the strong completeness and(strong) soundness of £Π. Notice that ZGG is an extension of £π in twosenses. Semantic extension: every interpretation aP of LG is an interpre-tation of LOG and vice versa. Moreover, for every formula φε LG, aP is amodel of φ in the sense of «£D. iff aP is a model of φ in the sense of ^.DG.Thus, for S, φ c LD, S\=φ in the sense of JjΠ iff S\=φ in the sense ofJLΠΏ. Consequently, the phrase "in the sense of . . . " is redundant—ifS, φ'Q LD, then the ambiguity of ζS)Fφ9 is inconsequential; if S, φ % LD,then 'S\=φ' can only mean (S\=φ in the sense of .£DD\ Deductive exten-sion: If Π is a proof in ZD then Π is a proof in ZGG. Hence, for S, φ cLD, if S\-φ in the sense of / D then S\-φ in the sense of ZDG. NOWsuppose S, φ c LD and S \-φ in the sense of JiΠΏ. By soundness S\=φ. Bystrong completeness of Z.G, S \-φ in the sense of -£D. Thus the phrase "inthe sense of . . . " is redundant here as above.

Let us introduce the usual definitions of &, v and =. For all φ and allψ, (φ&ψ) =~(φ ~>~Ψ), (φ v ψ) = (~φ Dψ) and (φ = ψ) = {{ψ Όψ)&(ψΏφ)).The usual rules of inference for &, v and = are derived rules in SCX andhence in JLU and ^.GG. The same holds for theorems involving the definedsymbols. In addition, we have

D3.0 \-U{φ & ψ) = {Πφ & Πψ)

h D ( ( ^ i & φ 2 & & φ n ) = (Πφ! & Πφ2 & . . . & Πφn)


D3.1 If ψ and ψh ψ2, . . . , ψn are modal formulas then\-Π(φ v ψ) = {Πφ v ψ)\-D(φi v ςί?2 v . . . v φn v ψλv ψ2v . . . v ψn) = (π(<pi V <ρ2 v . . . v φn)

v ψ i v ψ 2 v . . . vψ β )

Zλ?.(9 and Zλ3.2 will not be used immediately.

Lemma 3.1: If for every φε LDΠ ί tere exists φ'εLΠ such that \-φ = φ\

then JLΠU is strongly complete.

Proof. Assume the hypothesis and let S \=φ. For each ψεS choose a ψ'ε LΠsuch that \-ψ Ξψ' and let Sτ be the set of such formulas. Let φ1 be amember of LΠ such that \-φ = φ\ By soundness of -£DD, if \-η = τ]τ then\=η =η\ Thus Sτf=<?τ. By strong completeness of ZD, ST h<ρτ. Let Π be aproof in -£π of φ' from Sτ and let ψ[, ψ2, . . . , Ψn be the assumptions of Π.By ?z applications of (DI) we have a proof Πo of h(ψί D {ψ2 D . . . 3 (Ψi=>(^τ)) Let ψi, i//2, , Ψn be members of 5 provably equivalent to ψ[,ψ2, . . . , ψn and let Πi, Π2? . . . , Πw and Πw+i be respectively proofs of(ΦΊ D ψ[), (ψ2 ^ Ψ2), . . . , W'ra D ψ«) and ((/?τ D (p). These exist because ofthe "derived ru le" : {η = η') implies (η D r]τ) and (τ]τ D η).

ΠoΠxΠs, . . . , Un+1[Ψl[ψ2, , [*»^i^2, . . . , ΨniΨl D ( ^ D, . . . ,

The above is a proof of <ρ from ι//i, ψ2 . . . ψn Thus S h f The above proofis constructed as follows. The concatenation of Πo, Πi, Π2, . . . , and Πw+iis a proof having no open assumptions and having (ψλ D ψ[), (ψ2 D ψ2), . . . ,(ψw D ψ'n), (ψ[ D (ψ2 D, . . . , OK D ςp»), . . . ,- ) and ((/?τ z) ^) as open formu-las. Thus after ψl9 ψ2, . . . , and ψw have been introduced as assumptions,Ψh Ψl, , &nd i/4 can be added as open formulas by detachment ( D E ) .Now (ψ2 D {ψl D, . . . , (ψi D ^Oj' , ) can be added by detachment. Bycontinuing to detach the consequent of the last formula added we ultimiatelyadd φ\ Now we detach φ from (φ* D φ).

It remains only to show that every formula in LDD is provably equiva-lent to a formula in LD. For this we need "substitutivity of equivalence".For this, in turn, we need the following.

D3.2 \-((φ = ψ) D(~φ=~ψ))D3.3 \-{(φ = ψ ) D ({ηΏ φ) = (n 3 ψ))D3.4 h((φ =ψ)Ώ ((φ D η) = (ψ D η))D3.5 If \-(φ = ψ) then \-(Πφ = Πψ)

The first three of these are tautologies. D3.5 is essentially twoapplications of D1.3. Incidentally, it is not true that \=({φ = ψ) D (Πφ=Πψ)).

Theorem 3.2 ("Substitutivity of Equivalence"). If ψ occurs at least onceΓφλ

in η and 77./, is the result of replacing one particular occurrence of φ in η

I \Ψ\\by ψ then \-(φ = ψ) implies \-iη = η I . 1.


The proof of 3.2 involves a tedious induction on the subformulas of η inwhich φ occurs. The smallest of these is φ itself and \-(φ = ψ) implies\-(φ = ψ). Suppose that the theorem has been established for the nth largestsuch subformula. Then one of D3.2, D3.3, D3.4 or D3.5 will apply to estab-lish the theorem for the n + 1st largest. Since η is finite, the argumentterminates.

In order to proceed, we define the rank of a formula as the depth ofiterated boxes in it. Specifically, r(φ) = 0 for φε L, r(D<ρ) = r{φ) + 1,r(~φ) = r(φ), r(φ D ψ) = max(r(φ), r{ψ)). Notice that LΠ contains all andonly formulas of rank <1. Let us call φ an atom of ψ iff φ occurs in ψ andeither (1) φεC> and at least one occurrence of φ in ψ is not inside of aformula of the form Uη in ψ or (2) φ is itself a formula of the form Πη andat least one occurrence of φ is not (properly) inside of a formula of theform Πrf in ψ. E.g. if P, Q and R are in 0> then the only atom of ~D((DP DQ) D R) is G((DP D Q) =) R) and the atoms of ((P D Q ) D (D(R D DP) z> DP))are P, Q, D(R z> DP) and DP. Obviously, the rank of φ is the maximum ofthe ranks of the atoms of φ.

Since SC2 is complete we have the normal form theorems. In particu-lar, we have that every formula φ is provably equivalent to a formula φ' inconjunctive normal form. If we adopt the convention that a conjunctivenormal form of a tautology is the disjunction of the atoms and the negationsof the atoms of the tautology, then we have that every formula φ of LDD isprovably equivalent to a conjunction of one or more disjunctions of itsatoms and (or) their negations. Every such conjunctive normal form of φhas the same rank as φ.

Theorem 3.3 If r(φ) = n, n > 1 then there is a formula φ1 such that v(φ') - n

and hD</? = φ\

Proof\ Let r(φ) = n, n > 1. Then h p i ί Ί & ^ & , . & Ψn where ψl9

ψ2, . . . , and ψn are disjunctions of the atoms and (or) negations of atomsof φ. By D3.0. y-πφ = Πψλ & Πψ2 & . . . & Πψn. Without loss of generality,we may assume that each ψv = a± v a2 v . . . v an v β± v β2 v . . . v βm,where each α ; has the form a or ^a for azθ and where each βj has the formDβ or ^Dj3, Πβ being an atom of φ. By D3.1 \-Dψ ^(0(0?! v a2 v . . . v an) v(βi v β2 v . . . v βn)). Putting these together using substitutivity of equiva-lence we have \-Πφ = φ\ where φJ is a truth-functional combination of(1) the modal atoms of φ and (2) some new modal atoms of rank 1. Since φhas rank n > 1, φ' has the same rank.

Theorem 3.4 (Reduction Theorem): For every formula φε LDD there exists

a formula φ'ε LD such that \-φ = φ\

Proof: By induction on rank. For r(φ) = 0 and r(φ) - 1 use φ itself. Nowassume that for every formula φ of rank n there exists c^τεLD such that\-φ = φ\ Let v(φ) = n + 1. Then one or more atoms of φ are of rank n + 1and the rest are of rank less than n + 1. Using substitutivity of equivalenceand 3.3, replace every atom of rank n + 1 in φ by an equivalent formula ofrank n obtaining ψ of rank n such that \-φ =Ψ. By induction hypothesis\-ψ = ψ' where r(ψ') < 1. Thus \-φ = ψr where ψ'ε LD.


Theorem 3.5: ZDD is strongly complete.

Corollary 3.5: Kripke's semantics for S5 is compact.

Putting 3.5 together with 1.4 we have the strong bridge result for /DD.

4 Some further conclusions. .£DD is an extension of Kripke's system S5.The primary difference between the two systems stems from the fact thatKripke's system handles only "logical truth" whereas ^DD handles both"logical truth" and "valid arguments". Deductively, this means that \-φ isdefined in Kripke's system whereas \-φ and S\-φ are both defined in .£DD.Semantically, this means that \=φ is defined in Kripke's system, whereasboth \=φ and S\=φ are defined in JLΠU. In order to define \=φ it is sufficientto define interpretations of formulas. To define S \=φ it is much morenatural to define interpretations of the entire language. Our definitions of"interpretation of LDD" is obtained by replacing the words 'a formula' inKripke's definition of "interpretation of a formula" by the word 'LDD'.Since ^.DD is strongly sound and strongly complete, every other deductivesystem which has these two properties relative to the above semantics isessentially the same as the deductive system of JLΠΠ in the sense that forevery S, φ <Ξ LDD, φ is provable from S in the other system iff φ is prov-able fromS in ZDD.

Some logicians (e.g. Feys, [6] p. 6) have expressed a special interestin "superposed" or iterated modalistics. As a result of the above investi-gations we can assert that iterated modalities are completely dispensablewithin ^.DD. The reduction theorem (3.5) and the strong soundnesstheorem (1.4) imply that iterated modalities are dispensable in regard toexpressive power in the sense that for every formula φ containing iteratedmodalities there is another sentence φ having no iterated modalities andhaving exactly the same models as φ, i.e. \=φ = φ'- Moreover, the methodemployed in the proof of strong completeness also yields the result thatiterated modalities are dispensable in regard to deduction in the sense thatif S, φ contains iterated modalities and Sτ, φ" is a corresponding sem-antically equivalent set of formulas not containing iterated modalities, then5 \-φ implies that there is a proof of φ1 from S! none oί whose formulascontain iterated modalities.

As Theorem 2.5 is stated it holds for ZDD as well for ZD. AS aresult of this theorem we have that (1) Πφ is provable in ^ D iff φ is atautology. Naturally, this latter result does not hold in -£DD, e.g. if P andQ are in & then in ZDD we have ι-D(DP D D(Q D P)). Noticing that fact (1)compares provability of a rank 1 expression to a semantical property of arelated rank 0 expression, we can generalize it as follows. Let LDwbe the

sent of formulas of rank n. Thus LD0 = L, LDX = LD, and U LDW= LDD.n

Let JLun be the restriction of -£DD to LDwin the same sense that Z D is arestriction of ^.DD to LD. The method used in proving strong completenessfor -£DD from the strong completeness of JSλ can also be used to prove,for each n ^ 1, the strong completeness of -£DW+1 from the strong com-pleteness of -£DW. Strong soundness of -£DW, for all n, is a corollary of


strong soundness for /DD. Thus for all φ and all n, Πφ is provable inJLUn+i iff φ is logically true in the semantics of £Πn. In case n - 1 wehave fact (1) above.

The novelty of our approach to modal logic has been two-fold. In thefirst place, we emphasize the relation of logical consequence, whereas allprevious writers have restricted themselves to the consideration of logicaltruth. In our opinion, this restriction is unnecessary and, moreover, iteffectively blocks utilization of insights which have been achieved from thebroader viewpoint. In the second place, our deductive system was designedto embody semantical insights, i.e. the intuitions which would lead one toguess that a given argument is valid will also lead one to discover a proofof it in our system. This advantage saves an enormous amount of essen-tially useless manipulation.

REFERENCES

[1] Corcoran, John and George Weaver, ''Logical Consequence in Modal Logic:Natural deduction in S5," (abstract), The Journal of Symbolic Logic, v. 33 (1968),p. 639.

[2] Curry, Haskell B., Foundations of Mathematical Logic, New York (1963).

[3] Davis, Martin, Lecture Notes on Mathematical Logic, New York (1959); (mimeo-

graphed by Courant Institute, New York University).

[4] Feys, Robert, with Joseph Dopp, Modal Logics, Louvain (1965).

[5] Henkin, Leon, "The completeness of the first-order functional calculus," The

Journal of Symbolic Logic, v. 14 (1949), pp. 159-166.

[6] Kripke, Saul, "Semantical analysis of modal logic I , " Zeitschift fur mathe-matische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67-96.

[7] Leblanc, Hugues, Techniques of Deductive Inference, Englewood Cliffs, N.J.

(1966).

[8] Mendelson, Elliott, Introduction to Mathematical Logic, Princeton (1964).

University of PennsylvaniaPhiladelphia, Pennsylvania

LOGICAL CONSEQUENCE IN MODAL LOGIC: NATURAL DEDUCTION IN S5 by JOHN CORCORAN and GEORGE WEAVER

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